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Landmark-Matching Transformation with Large Deformation Via n-dimensional Quasi-conformal Maps

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Abstract

We propose a new method to obtain landmark-matching transformations between n-dimensional Euclidean spaces with large deformations. Given a set of feature correspondences, our algorithm searches for an optimal folding-free mapping that satisfies the prescribed landmark constraints. The standard conformality distortion defined for mappings between 2-dimensional spaces is first generalized to the n-dimensional conformality distortion K(f) for a mapping f between n-dimensional Euclidean spaces \((n \ge 3)\). We then propose a variational model involving K(f) to tackle the landmark-matching problem in higher dimensional spaces. The generalized conformality term K(f) enforces the bijectivity of the optimized mapping and minimizes its local geometric distortions even with large deformations. Another challenge is the high computational cost of the proposed model. To tackle this, we have also proposed a numerical method to solve the optimization problem more efficiently. Alternating direction method with multiplier is applied to split the optimization problem into two subproblems. Preconditioned conjugate gradient method with multi-grid preconditioner is applied to solve one of the sub-problems, while a fixed-point iteration is proposed to solve another subproblem. Experiments have been carried out on both synthetic examples and lung CT images to compute the diffeomorphic landmark-matching transformation with different landmark constraints. Results show the efficacy of our proposed model to obtain a folding-free landmark-matching transformation between n-dimensional spaces with large deformations.

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Acknowledgments

The authors acknowledge the freely available lung CT data from the Deformable Image722 Registration Laboratory (www.dir-lab.com). Lok Ming Lui is supported by RGC GRF (CUHK Project ID: 2130363; Reference: 402413).

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Authors and Affiliations

  1. Room E18-301A, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Yin Tat Lee

  2. Lady Shaw Building Room 222B, Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Ka Chun Lam

  3. Lady Shaw Building Room 207, Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Lok Ming Lui

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  1. Yin Tat Lee
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Correspondence to Lok Ming Lui.

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Lee, Y.T., Lam, K.C. & Lui, L.M. Landmark-Matching Transformation with Large Deformation Via n-dimensional Quasi-conformal Maps. J Sci Comput 67, 926–954 (2016). https://doi.org/10.1007/s10915-015-0113-5

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